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GRE Subject Test: Math : Derivatives & Integrals

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Example Questions

Example Question #1 : Trigonometric Integrals

Fnd the derivative of tan(x) with respect to x or

$\frac{\mathrm{d} }{\mathrm{d} x} tan(x)dx$ $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} tan(x)dx$

Possible Answers:

$\displaystyle cos^2(x)$

Derivative cannot be found

sec(x) $\displaystyle sec(x)$

cos(x) $\displaystyle cos(x)$

$\displaystyle sec^2(x)$

Correct answer:

$\displaystyle sec^2(x)$

Explanation:

The is one of the trigonometric integrals that must be memorized.

$\frac{\mathrm{d} }{\mathrm{d} x} tan(x)dx=sec^2(x)$ $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} tan(x)dx=sec^2(x)$

Other common trig derivatives that should be memorized are:

$\frac{\mathrm{d} }{\mathrm{d} x} sin(x)dx=cos(x)$ $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} sin(x)dx=cos(x)$

$\frac{\mathrm{d} }{\mathrm{d} x} cos(x)dx=-sin(x)$ $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} cos(x)dx=-sin(x)$

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Example Question #2 : Trigonometric Integrals

Evaluate:

$\int_{0}^{\pi /3}\left ( \frac{cosx}{2}\right )dx$ $\displaystyle \int_{0}^{\pi /3}\left ( \frac{cosx}{2}\right )dx$

Possible Answers:

$\frac{1}{4}$ $\displaystyle \frac{1}{4}$

$\frac{\pi}{3}$ $\displaystyle \frac{\pi}{3}$

$\frac{\sqrt3}{2}$ $\displaystyle \frac{\sqrt3}{2}$

$\frac{\sqrt3}{4}$ $\displaystyle \frac{\sqrt3}{4}$

$\frac{\sqrt3-4}{4}$ $\displaystyle \frac{\sqrt3-4}{4}$

Correct answer:

$\frac{\sqrt3}{4}$ $\displaystyle \frac{\sqrt3}{4}$

Explanation:

1) The 1/2 is a constant, and so is pulled out front.

2) The integral of cos(x) is sin(x), by definition.

3) Writing the limits for evaluation:

$\frac{1}{2}sin(x)|^{\pi/3}_{0} = \frac{1}{2}\left [ sin(\pi/3)-sin(0)\right ]$ $\displaystyle \frac{1}{2}sin(x)|^{\pi/3}_{0} = \frac{1}{2}\left [ sin(\pi/3)-sin(0)\right ]$

4) Using the unit circle, $sin(\pi/3)$ $\displaystyle sin(\pi/3)$ $=\sqrt3/2$ $\displaystyle =\sqrt3/2$, and $\displaystyle sin(0)=0$.

5)Simplifying:

$\frac{1}{2}\left [ \frac{\sqrt3}{2}-0\right ]=\frac{\sqrt3}{4}$ $\displaystyle \frac{1}{2}\left [ \frac{\sqrt3}{2}-0\right ]=\frac{\sqrt3}{4}$

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GRE Subject Test: Math : Derivatives & Integrals

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Trigonometric Integrals

Example Question #2 : Trigonometric Integrals

All GRE Subject Test: Math Resources

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